Eamcet - Maths - Theory Of Equations

If the roots of the equation 4x3-12x2+11 x + k = 0 are in arithmetic progression, then k is equal to :

A.  -3

B.  1

C.  2

D.  3

If f(x) is a polynomial of degree n with rational coefficients and 1+2i, 2-√3 and 5 are three roots of       f(x) = 0, then the least value of n is :

A.  5

B.  4

C.  3

D.  6

If cos 2x = (√2 + 1)(cos x - 1/√2) , cos x ≠ 1/2 then x belongs to

A.  {2nπ ± π/6 : n Є Z }

B.  {2nπ ±π/3 : n Є Z }

C.  {2nπ ±π/2 : n Є Z }

D.  {2nπ ± π/4 : n Є Z }

The sum of the fourth powers of the roots of the equation x5+px3+qx2+s=0 is

A.  2p2

B.  3p2

C.  2p

D.  3p

If the roots of x4+5x3-30x2-40x+64=0 are in G.P then the roots are

A.   1,-2,4,-8

B.  ±1, 2,3

C.  ±2i, 2 3

D.  -3/2,-1/3,2±√3

If A,B,C are the remainders of x3-3x2-x+5,3x4-x3+2x2-2x-4,2x5-3x4+5x3-7x2+3x-4 when divided by x+1,x+2,x-2 respectively then the ascending order of A,B,C is

A.  A,B,C

B.  B,C,A

C.  A,C,B

D.  B,A,C

If α and β are different values of x satisfying a cos x+b sin x=c then tan (α+β/2) =

A.  a+b

B.  a-b

C.   a/b

D.  b/a

If x2+p1x+q1=0,x2+p2x+q2=0,x2+p3x+q3=0 has a common root,then p12+p22+p32+4(q1+q2+q3)=

A.  2(p1p2+p2p3+p3p1)

B.  (p2p1+q2p3+q3p1)

C.  2(q1p2+q2p3+p3q1)

D.  none

The equation whose roots are multiplied by 3 of those of 2x2+3x-1=0 is

A.  2x2+9x-9=0

B.  2x2-7x+6=0

C.  2x2+5x+7=0

D.  3x2+5x-7=0

The roots of x3-3x2+4=0,when there is a multiple root,are

A.  1/2,1,-3

B.  1/3,1/3,1

C.  6,4,-1

D.  2,2,-1

The equations whose roots are exceed by 2 than those of x4+x3-10x2+4x+24=0 is

A.  x4-7x3+8x2+24x-16=0

B.  x5-7x3+12x2-7x=0

C.  x4+7x3-11x2-144x-208=0

D.  x5+11x4+45x3+81x2+50x-6=0

If 2,-2,4 are the roots of ax3+bx2+cx+d=0 then the roots of 8ax3+4bx2+2cx+d=0 are

A.  2,-2,4

B.  1/2,-1/2,1/4

C.  1,-1,2

D.  4,-4,8

If α+β+γ=1,α2+β2+γ2=2 and α3+β3+γ3=3,then α5+β5+γ5=

A.  6

B.  18

C.  36

D.  45

If the equations x2-x-p=0 and x2+2px-12=0 have a common root,then that root is

A.  i

B.  p+2

C.  2

D.  cannot be determined

If a=cos 2π/7+i sin 2π/7,α=a+a2+a4 and β=a3+a5+a6 then α,β are the roots of the equation

A.  x2+x+1=0

B.  x2+x+2=0

C.  x2+2x+2=0

D.  x2+2x+3=0

The values of the parameters a for which the quadratic equations (1-2a)x2-6ax-1=0 and ax2-x+1=0 have at least one root in common are

A.  0,1/2

B.  1/2,2/9

C.  2/9

D.  0,1/2,2/9

The roots of x4-5x3+3x2+19x-30=0 are

A.  -2,3,3±i

B.  -2,3,2±3i

C.  -2,3,2±i

D.  -2,3,3±2i

If α,β are the roots of 8x2-3x+27=0 then the value of (α2/β)1/3+(β2/α)1/3 is

A.  1/3

B.  1/4

C.  7/2

D.  4

The roots of x3+x2-2x-2=0 are

A.  -1,±√2

B.  0,1,2

C.  -2,-1,1

D.  1,±i

The transformed equation of x3-4x2+1/4x-1/9=0,by eliminating fractional coefficients is

A.  y3+15y2+52y-36=0

B.  y4-24y2+65y-55=0

C.  4x4-2x3+6x2-3x-1=0

D.  y3-24y2+9y-24=0

The remainder when 2x5-3x4+5x3-3 x2+7x-9 with x2-x-3 is

A.  33x+4

B.  41x+3

C.  41x+44

D.  33x-4

The equations whose roots are diminish by 3 than those of x4-5x3-20x2+3x+17=0 is

A.  x4-9x3+20x2=0

B.  x5-7x3+12x2-7x=0

C.  x4+7x3-11x2-144x-208=0

D.  x5+11x4+45x3+81x2+50x-6=0

If α,β,γ are the roots of x3+x2+x+1=0 then α3+β3+γ3=

A.  1

B.  -1

C.  2

D.  -2

If one root of the equation ix2-2(1+i)x+(2-i)=0 is 2-I,then the other root is

A.  -i

B.  2+i

C.  i

D.  2-i

Match the following. Equation Roots I.x3-3x2-16x+48 =0 a)6,4,-1 II.x3-7x2+14x-8=0 b)1,1/3,1/5 III.15x3-23x2-9x-1=0 c)1,2,4 IV. x3-9x2+14x+24=0 d)4,-4,3

A.  c, d, a, b

B.  d ,c, b, a

C.  c, a, b, d

D.  c, b, a, d

If α,β,γ are the roots of x3+3px+q=0 then the equation whose roots are α+1/β+γ–α,β+1/γ+α–β and γ+1/α+β–γ is

A.  8y3+12y3+(6+6p) y + 1 +3p –q=0

B.  8qy3-12(q+ p)y2 + 6(q-2p) y + (q+3p-1)=0

C.  8qy3+12(q+ p)y2 + 6(q-2p) y + (q+3p-1)=0

D.  8qy3-12(q+ p)y2 -6(q-2p) y + (q+3p-1)=0

If the quadratic equations ax2+2cx+b=0 and ax2+2bx+c=0 (b≠c) have a common root,then a+4b+4c=

A.  -2

B.  -1

C.  0

D.  1

If α,β,γ are the roots of x3+2x2-3x-1=0,then the value of α-4+β-4+γ-4 is

A.  123

B.  36

C.  149

D.  795

If A12/x-a1+A22/x-a2+.... Ak2/x-a k=m and ai,AI,m are rational then the equation has

A.  no imaginary roots

B.  no positive roots

C.  no negative roots

D.  no real roots

If the sum of two of the roots of 4x3+16x2-9x-36=0 is zero then the roots are

A.  ±√5, 1+i

B.  3/2,-3/2,-4

C.  -1/2, 1/2,-1/5

D.  ±√2, √5